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## Tic-Tac-Toe game variations (February 12, 2004)

Most of us are familiar with Tic-Tac-Toe: players take turns placing X's and O's in a 3x3 square and the winner is the person who gets 3-in-a-row.

Of course, it is possible that there is no winner -- the game ends in a draw!

There are two variations on Tic-Tac-Toe:

## Achi

This is a game played by the Asante people of Ghana, West Africa *. It is played on a board like the one shown below:

## Rules

Each player starts with four counters (like X's and O's) and takes turns placing them on the board as in Tic-Tac-Toe, with the goal of getting a 3-in-a-row. However, if the game is a draw after each has played their four counters, they take turns sliding a counter along the lines into the space left empty. The winner is the first player to get 3-in-a-row.

For example, if the players have played their counters in the following way (the first player in red and the second player in blue),

then the first player then must slide the counter in the bottom right corner into the open (white) space, and then the second player slides one of his/her pieces, etc. (Which piece would you slide if you were the second player?)

## Questions

1. If no one won after placing the counters, can the players always slide a counter until someone wins?
2. Does either player have an advantage? If so, why?
3. Do they have a strategy that guarantees they can win, no matter what the other player does? If so, what does such a strategy look like?

## Rules

Think of three Tic-Tac-Toe boards stacked one on top of another.

You can draw this on paper by placing the boards side by side on a piece of paper and thinking of them as top, middle and bottom.) The goal is still to be the first to get 3-in-a-row, but now the 3-in-a-row can be on any of the three levels, or between levels. For example, the two pictures below show 3-in-a-rows that use all three levels.

 Top Middle Bottom

 Top Middle Bottom

## Questions

1. How many different 3-in-a-row's are there?
2. Can the game end in a draw?
3. What are some ways in which you can force a win? Do you have an advantage as the first or second player?